Mathematical competence is acquired through
a careful balance of conceptual understanding, problem
solving, and logical proofs. The drill of skills should
receive decreased attention and mathematical proofs should
not be used as the most important method of fostering
understanding. Rather, more emphasis should be placed
on conceptual clarity and creative problem solving in
order to develop a mature mathematical competence.

In this manner, the teacher must be
simultaneously a transmitter of the knowledge and a
coach, in order to motivate students and their problem
solving skills. Effective mathematics teaching should
involve active student participation during class time
as a very important component of the teaching strategy.

As implied earlier in my philosophy
of education, problem solving is especially important
for the students studying mathematics. The teacher must
use problem solving strategies (i.e. the scientific
method) that require persistence, the ability to recognize
wrong assumptions, analyze a situation through the use
of trial and error, modeling graphing, and drawing conclusions.

Further, through the guidance of the
mathematics teacher, students must be taught to apply
inductive and deductive reasoning techniques to construct
mathematical arguments. They must be able to develop
conjectures on the basis of intuition and test these
conjectures by using logic and deductive and inductive
proof. Additionally, students must be taught to judge
the validity of mathematical arguments.

Finally, the mathematics teacher should
make consorted efforts to ensure that students are taught
the skills necessary to read, write, and speak mathematics,
and relate and connect mathematics to real-world applications
and across the curriculum. In this manner, the mathematics
teacher should demonstrate to students that mathematics
is a growing subject interrelated with many other disciplines.